Search Previous Year Questions

Shortcut Trick

Apply the Successive Discount formula: a + b − (a × b)/100.

Given: a = 30% and b = 10%.

Net Discount % = 30 + 10 − (30 × 10)/100

Net Discount % = 40 − 3 = 37%

∴ The correct answer is 37%.

Alternate Method

Given: Original Price = 1,500, First Discount = 30%, Second Discount = 10%.

⇒ First discount amount = 30% of 1,500 = (30/100) × 1500 = 450

⇒ Price after 1^st discount = 1500 − 450 = 1050

⇒ Second discount amount = 10% of 1,050 = (10/100) × 1050 = 105

⇒ Final Selling Price = 1050 − 105 = 945

⇒ Total Discount = 1500 − 945 = 555

⇒ Equivalent Discount % = (Total Discount / Original Price) × 100

⇒ Equivalent Discount % = (555 / 1500) × 100 = 37%

∴ The correct answer is 37%.

Additional Information

Net Discount for Three Rates

For three successive discounts a%, b%, and c%, the net discount is calculated by first finding the net of a and b, then applying the formula to that result and c.

Marked Price (MP) and Selling Price (SP)

Discount is always computed on the Marked Price. SP = MP × (100 − D₁%)/100 × (100 − D₂%)/100.

Multiplying Factor Method

Final Price = Initial Price × 0.70 × 0.90. The overall multiplier is 0.63, which implies a (1 − 0.63) × 100 = 37% discount.

Shortcut Trick

Interest earned in 1 year = 3600 ÷ 4 = ₹900

Rate of Interest % = (Annual Interest ÷ Principal) × 100

Rate % = (900 ÷ 15000) × 100 = 6%

∴ The correct answer is 6%.

Alternate Method

Given:

Principal (P) = ₹15,000

Time (T) = 4 years

Simple Interest (SI) = ₹3,600

Formula Used:

Simple Interest (SI) = (P × R × T) ÷ 100

Rate (R) = (SI × 100) ÷ (P × T)

Calculations:

⇒ R = (3600 × 100) ÷ (15000 × 4)

⇒ R = 360000 ÷ 60000

⇒ R = 6

∴ The correct answer is 6%.

Additional Information

Amount Calculation

The total sum to be repaid after the time period is the Amount, calculated as Amount = Principal + Simple Interest.

Time and Rate Relationship

For a fixed interest amount and principal, the Rate of Interest is inversely proportional to the Time (R ∝ 1/T).

Constant Annual Interest

In Simple Interest, the interest earned remains the same for each year of the time period.

Shortcut Trick

Pick Assumed Mean (A) = 28. Find the mid-point (x) for each class interval: 4, 12, 20, 28, 36, 44.

Calculate deviations d = x − A: −24, −16, −8, 0, 8, 16.

Calculate Σf×d = 4(−24) + 6(−16) + 7(−8) + 3(0) + 12(8) + 8(16) = −96 − 96 − 56 + 0 + 96 + 128 = −24.

Mean = A + (Σfd ÷ Σf) = 28 + (−24 ÷ 40) = 28 − 0.6 = 27.4.

∴ The correct answer is 27.4.

Alternate Method

Given: Frequencies and class intervals for a dataset.

Formula Used: Direct Method Mean (μ) = Σ(f_i × x_i) ÷ Σf_i

Step 1: Calculate Mid-points (x_i) and Products (f_ix_i):

⇒ (0-8): x₁ = 4, f₁x₁ = 4 × 4 = 16

⇒ (8-16): x₂ = 12, f₂x₂ = 6 × 12 = 72

⇒ (16-24): x₃ = 20, f₃x₃ = 7 × 20 = 140

⇒ (24-32): x₄ = 28, f₄x₄ = 3 × 28 = 84

⇒ (32-40): x₅ = 36, f₅x₅ = 12 × 36 = 432

⇒ (40-48): x₆ = 44, f₆x₆ = 8 × 44 = 352

Step 2: Calculate Total Frequency and Sum of Products:

⇒ Σf_i = 4 + 6 + 7 + 3 + 12 + 8 = 40

⇒ Σf_ix_i = 16 + 72 + 140 + 84 + 432 + 352 = 1096

Step 3: Calculate the Mean:

⇒ Mean = 1096 ÷ 40

⇒ Mean = 27.4

∴ The correct answer is 27.4.

Additional Information

Direct Method for Mean

It is the simplest method where Mean = Σf_ix_i ÷ Σf_i, used when values of x_i and f_i are small.

Assumed Mean Method

Used to simplify calculations by choosing a central value A. Mean = A + (Σf_id_i ÷ Σf_i), where d_i = x_i − A.

Step-Deviation Method

Used when class sizes (h) are uniform. Mean = A + [(Σf_iu_i ÷ Σf_i) × h], where u_i = (x_i − A) ÷ h.

Empirical Relationship

The connection between the three measures of central tendency: 3 × Median = Mode + 2 × Mean.

Given:

((³√4096) ÷ (⁴√10000)) × (5/48) × 186

Formula used:

Cube root: ³√x = x^1/3

Fourth root: ⁴√x = x^1/4

Calculations:

Evaluate the roots:

³√4096 = 16

⁴√10000 = 10

Substitute values and simplify the expression:

⇒ (16 ÷ 10) × (5 ÷ 48) × 186

⇒ (8/5) × (5/48) × 186

⇒ (8/48) × 186

⇒ (1/6) × 186

⇒ 31

∴ The correct answer is option (4).

Shortcut Trick

Ratio of Marked Price to Reduced Price = 500 : 450 = 10 : 9.

Decrease in price = 10 − 9 = 1 unit.

Percentage decrease = (Decrease ÷ Original Price) × 100 = (1 ÷ 10) × 100 = 10%.

∴ The correct answer is 10%.

Alternate Method

Given:

Marked Price (MP) = ₹500

Reduced Price (SP) = ₹450

Formula Used:

Percentage Decrease = [(Original Value − New Value) ÷ Original Value] × 100

Calculations:

⇒ Total reduction in price = 500 − 450 = ₹50

⇒ Percentage decrease = (50 ÷ 500) × 100

⇒ Percentage decrease = (1 ÷ 10) × 100

⇒ Percentage decrease = 10%

∴ The correct answer is 10%.

Additional Information

Discount and Marked Price

The amount deducted from the Marked Price is called the Discount. Discount % is always calculated on the Marked Price: Discount % = (Discount ÷ MP) × 100.

Successive Discounts

When two or more discounts are applied one after another, the effective discount is calculated using the formula: x + y − (x × y ÷ 100).

Selling Price Calculation

The final price after the discount is the Selling Price (SP). SP = MP × (100 − Discount %) ÷ 100.

Shortcut Trick

Co-prime numbers are pairs that have no common factor other than 1 (HCF = 1).

Analyze options: 27 and 36 are divisible by 9; 44 and 99 are divisible by 11; 42 and 56 are divisible by 14.

For 25 and 36, their factors do not overlap: 25 (5 × 5) and 36 (2 × 2 × 3 × 3).

∴ The correct answer is 25 and 36.

Alternate Method

Given: A set of number pairs to identify the co-prime pair.

Definition: Two numbers are co-prime if their Highest Common Factor (HCF) is 1.

Calculation for all options:

⇒ Option 1: HCF(27, 36) → Factors of 27 = {1, 3, 9, 27}; Factors of 36 = {1, 2, 3, 4, 6, 9, ...} ⇒ HCF = 9

⇒ Option 2: HCF(44, 99) → Factors of 44 = {1, 2, 4, 11, 22, 44}; Factors of 99 = {1, 3, 9, 11, 33, 99} ⇒ HCF = 11

⇒ Option 3: HCF(25, 36) → Factors of 25 = {1, 5, 25}; Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36} ⇒ HCF = 1

⇒ Option 4: HCF(42, 56) → Factors of 42 = {1, 2, 3, 6, 7, 14, ...}; Factors of 56 = {1, 2, 4, 7, 8, 14, ...} ⇒ HCF = 14

∴ The correct answer is 25 and 36.

Additional Information

Properties of Co-prime Numbers

Any two prime numbers are always co-prime to each other. Also, any two consecutive numbers are always co-prime (e.g., 9 and 10).

HCF and LCM Relation

For a pair of co-prime numbers a and b, their HCF = 1 and their LCM = a × b.

Perfect Squares

Two perfect squares can be co-prime if their square roots do not share any common factors (e.g., 25 = 5² and 36 = 6²).

Shortcut Trick

The overall profit is the product of individual multipliers for markup, discount, and weight fraud.

Net Multiplier = (1 + 24/100) × (1 − 18/100) × (1000 ÷ 800)

Net Multiplier = 1.24 × 0.82 × 1.25

Net Multiplier = 1.0168 × 1.25 = 1.271

Profit % = (1.271 − 1) × 100 = 27.1%

∴ The correct answer is 27.1%.

Alternate Method

Given:

Markup = 24% above CP | Discount = 18% | Actual weight = 800 g | Declared weight = 1 kg (1000 g).

Formula Used:

Overall Profit % = [(Total SP − Total CP) ÷ Total CP] × 100

Calculations:

⇒ Let the Cost Price (CP) of 1000 g of sugar be ₹1000 (so ₹1 per gram).

⇒ Marked Price (MP) = 1000 + 24% of 1000 = ₹1240.

⇒ Selling Price (SP) = 1240 − 18% of 1240 = 1240 − 223.2 = ₹1016.8.

⇒ Since the shopkeeper gives only 800 g, his actual CP = ₹800.

⇒ Total Profit = SP − Actual CP = 1016.8 − 800 = ₹216.8.

⇒ Overall Profit % = (216.8 ÷ 800) × 100 = 27.1%.

∴ The correct answer is 27.1%.

Additional Information

Successive Percentage Change

When multiple changes occur, the net effect is calculated using multipliers: Net Factor = (1 ± r₁/100) × (1 ± r₂/100) × ...

Dishonest Dealer Rule

Profit % due to weight fraud = [(True Weight − False Weight) ÷ False Weight] × 100, assuming sale is at CP.

CP, MP, and SP Relationship

The direct ratio is given by (100 + Profit%) ÷ (100 − Discount%) = Marked Price ÷ Cost Price.

Shortcut Trick

Net Rate Difference per year = 6.5% − 3.5% = 3%.

Total Interest Difference for 2 years = 3% × 2 = 6%.

Required Difference = 6% of ₹2,60,000.

⇒ 2,60,000 × (6 ÷ 100) = 2,600 × 6 = 15,600.

∴ The correct answer is ₹15,600.

Alternate Method

Given:

Principal (P) for Bank A and Bank B = ₹2,60,000 each.

Interest Rate (R₁) for Bank A = 3.5% per annum.

Interest Rate (R₂) for Bank B = 6.5% per annum.

Time period (T) = 2 years.

Formula Used:

Simple Interest (SI) = (Principal × Rate × Time) ÷ 100

Calculations:

⇒ Interest paid to Bank A (SI_A) = (2,60,000 × 3.5 × 2) ÷ 100

⇒ SI_A = 2,600 × 7 = 18,200

⇒ Interest paid to Bank B (SI_B) = (2,60,000 × 6.5 × 2) ÷ 100

⇒ SI_B = 2,600 × 13 = 33,800

⇒ Positive Difference = SI_B − SI_A

⇒ Difference = 33,800 − 18,200 = 15,600

According to the calculation, the interest difference is ₹15,600 which corresponds to option (2).

∴ The correct answer is ₹15,600.

Additional Information

Simple Interest Principle

Simple interest is calculated only on the initial Principal amount. The formula is SI = (P × R × T) ÷ 100, where P is Principal, R is annual rate, and T is time in years.

Interest Difference Shortcut

When Principal and Time are constant, the difference in interest is directly proportional to the difference in rates: Difference = [P × T × (R₂ − R₁)] ÷ 100.

Total Amount

The total amount (A) to be repaid after the time period is the sum of the Principal and the Interest: A = P + SI.

Shortcut Trick

Area of rectangular field = 55 × 70 = 3850 m².

As per the problem, Area of circular field = πr² = 3850 m².

⇒ (22/7) × r² = 3850 ⇒ r² = 175 × 7 = 1225 ⇒ r = 35 m.

Perimeter (Circumference) = 2πr = 2 × (22/7) × 35 = 220 m.

∴ The correct answer is 220 m.

Alternate Method

Given:

Length of the rectangular field (L) = 70 m

Breadth of the rectangular field (B) = 55 m

Formula Used:

Area of rectangle = Length × Breadth

Area of circular field = π × r²

Perimeter (Circumference) of circle = 2 × π × r

Calculations:

⇒ Area of rectangular field = 70 × 55 = 3850 m²

⇒ Area of circular field = 3850 m²

⇒ πr² = 3850

⇒ (22/7) × r² = 3850

⇒ r² = (3850 × 7) ÷ 22

⇒ r² = 175 × 7 = 1225

⇒ Radius (r) = √1225 = 35 m

⇒ Perimeter = 2 × (22/7) × 35

⇒ Perimeter = 2 × 22 × 5 = 220 m

∴ The correct answer is 220 m.

Additional Information

Circle Properties

The ratio of the circumference to the diameter of any circle is always constant, denoted by π (≈ 22/7 or 3.14).

Comparison of Perimeter

For a given area, a circle has the shortest perimeter compared to any other plane figure like a rectangle or square.

Unit Conversion

Ensure all dimensions are in the same units (e.g., meters) before equating areas or calculating perimeters.

Shortcut Trick

For a number to be divisible by 9, the sum of its digits must be a multiple of 9.

Sum of digits for 7KM93 = 7 + K + M + 9 + 3 = 19 + K + M.

Possible sums are 27 or 36, giving K + M = 8 or K + M = 17.

To minimize 4K² + 2M, keep K as small as possible because it has a higher power and coefficient.

Let K = 0, then M = 8 ⇒ 4(0)² + 2(8) = 16.

∴ The correct answer is 16.

Alternate Method

Given: A five-digit number 7KM93 is divisible by 9. We need to find the least value of the expression 4K² + 2M.

Formula Used: Sum of digits for divisibility by 9.

⇒ Sum of digits = 7 + K + M + 9 + 3 = 19 + K + M

⇒ Since the number is divisible by 9, 19 + K + M = 27 or 36

⇒ Case 1: K + M = 27 − 19 = 8. Possible (K, M) are (0, 8), (1, 7), (2, 6), ..., (8, 0).

⇒ For K = 0, M = 8: 4(0)² + 2(8) = 16.

⇒ For K = 1, M = 7: 4(1)² + 2(7) = 4 + 14 = 18.

⇒ Case 2: K + M = 36 − 19 = 17. Possible (K, M) are (8, 9) or (9, 8).

⇒ For K = 8, M = 9: 4(8)² + 2(9) = 4(64) + 18 = 256 + 18 = 274.

⇒ Comparing all values, the least value is 16.

∴ The correct answer is 16.

Additional Information

Divisibility by 3

A number is divisible by 3 if the sum of all its digits is divisible by 3.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd positions and even positions is either 0 or a multiple of 11.

Divisibility by 8

A number is divisible by 8 if its last three digits form a number that is divisible by 8.